# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Nonuniform exponential contractions and Lyapunov sequences. (English) Zbl 1173.37029
The paper deals with a nonautonomous dynamics with discrete time obtained from the product of linear operators. For this dynamics, it is shown that a nonuniform exponential contraction can be completely characterized in terms of so-called strict Lyapunov sequences. Nonuniform exponential contractions include as a very particular case the uniform exponential contractions that correspond to have a uniform asymptotic stability of the dynamics. The paper also presents “inverse theorems” that give explicitly strict Lyapunov sequences for each nonuniform exponential contraction. Essentially, the Lyapunov sequences are obtained in terms of what are usually called Lyapunov norms, that is, norms with respect to which the behavior of a nonuniform exponential contraction becomes uniform. It is also shown how the characterization of nonuniform exponential contractions in terms of quadratic Lyapunov sequences can be used to establish in a simple manner the persistence of the asymptotic stability of a nonuniform exponential contraction under sufficiently small linear or nonlinear perturbations. In addition, an appropriate version of the results obtained is described in the context of ergodic theory showing that the existence of an eventually strict Lyapunov function implies that all Lyapunov exponents are negative almost everywhere.
##### MSC:
 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 93D30 Scalar and vector Lyapunov functions
##### References:
 [1] Barreira, L.; Pesin, Ya.: Lyapunov exponents and smooth ergodic theory, Univ. lecture ser. 23 (2002) [2] Barreira, L.; Pesin, Ya.: Nonuniform hyperbolicity, Encyclopedia math. Appl. 115 (2007) [3] Barreira, L.; Schmeling, J.: Sets of ”non-typical” points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116, 29-70 (2000) · Zbl 0988.37029 · doi:10.1007/BF02773211 [4] Barreira, L.; Valls, C.: Stability of nonautonomous differential equations in Hilbert spaces, J. differential equations 217, 204-248 (2005) · Zbl 1088.34053 · doi:10.1016/j.jde.2005.05.008 [5] Barreira, L.; Valls, C.: Stability theory and Lyapunov regularity, J. differential equations 232, 675-701 (2007) · Zbl 1184.37022 · doi:10.1016/j.jde.2006.09.021 [6] Bhatia, N.; Szegö, G.: Stability theory of dynamical systems, Grundlehren math. Wiss. 161 (1970) · Zbl 0213.10904 [7] Conley, C.; Miller, R.: Asymptotic stability without uniform stability: almost periodic coefficients, J. differential equations 1, 333-336 (1965) · Zbl 0145.11401 · doi:10.1016/0022-0396(65)90011-2 [8] Eliasson, L.: Almost reducibility of linear quasi-periodic systems, Proc. sympos. Pure math. 69, 679-705 (2001) · Zbl 1015.34028 [9] W. Hahn, The present state of Lyapunov’s direct method, in: Nonlinear Problems, Madison, 1962, 1963, pp. 195 – 205 · Zbl 0111.28403 [10] Hahn, W.: Stability of motion, Grundlehren math. Wiss. 138 (1967) · Zbl 0189.38503 [11] Katok, A.; Burns, K.: Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems, Ergodic theory dynam. Systems 14, 757-785 (1994) · Zbl 0816.58029 · doi:10.1017/S0143385700008142 [12] Lasalle, J.; Lefschetz, S.: Stability by Liapunov’s direct method, with applications, Math. sci. Eng. 4 (1961) · Zbl 0098.06102 [13] Lewowicz, J.: Lyapunov functions and topological stability, J. differential equations 38, 192-209 (1980) · Zbl 0418.58012 · doi:10.1016/0022-0396(80)90004-2 [14] Lewowicz, J.: Lyapunov functions and stability of geodesic flows, Lecture notes in math. 1007, 463-479 (1983) · Zbl 0514.58031 [15] Lyapunov, A.: The general problem of the stability of motion, (1992) · Zbl 0786.70001 [16] Markarian, R.: Non-uniformly hyperbolic billiards, Ann. fac. Sci. Toulouse math. (5) 3, 1207-1239 (1994) [17] Oseledets, V.: A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Moscow math. Soc. 19, 197-221 (1968) · Zbl 0236.93034 [18] Sacker, R.; Sell, G.: Existence of dichotomies and invariant splittings for linear differential systems. I, J. differential equations 15, 429-458 (1974) · Zbl 0294.58008 · doi:10.1016/0022-0396(74)90067-9 [19] Wojtkowski, M.: Invariant families of cones and Lyapunov exponents, Ergodic theory dynam. Systems 5, 145-161 (1985) · Zbl 0578.58033 · doi:10.1017/S0143385700002807 [20] Wojtkowski, M.: Monotonicity, J-algebra of potapov and Lyapunov exponents, Proc. sympos. Pure math. 69, 499-521 (2001) · Zbl 1013.37028